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BUREAU OF MINES 
INFORMATION CIRCULAR/1990 




Utilizing Mechanical Linear 
Transducers for the 
Determination of a Mining 
Machine's Position and 
Heading: The Concept 

By Christopher C Jobes 




.P.NT O^ ' 



a 



80 



V YEARS 



(ft 

3 
O 



& 



U.S. BUREAU OF MINES 
1910-1990 

THE MINERALS SOURCE 



Mission: As the Nation's principal conservation 
agency, the Department of the Interior has respon- 
sibility for most of our nationally-owned public 
lands and natural and cultural resources. This 
includes fostering wise use of our land and water 
resources, protecting our fish and wildlife, pre- 
serving the environmental and cultural values of 
our national parks and historical places, and pro- 
viding for the enjoyment of life through outdoor 
recreation. The Department assesses our energy 
and mineral resources and works to assure that 
their development is in the best interests of all 
our people. The Department also promotes the 
goals of the Take Pride in America campaign by 
encouraging stewardship and citizen responsibil- 
ity for the public lands and promoting citizen par- 
ticipation in their care. The Department also has 
a major responsibility for American Indian reser- 
vation communities and for people who live in 
Island Territories under U.S. Administration. 



/ Information Circular 9254 



Utilizing Mechanical Linear 
Transducers for the 
Determination of a Mining 
Machine's Position and 
Heading: The Concept 



By Christopher C Jobes 



UNITED STATES DEPARTMENT OF THE INTERIOR 
Manuel Lujan, Jr., Secretary 

BUREAU OF MINES 
T S Ary, Director 



<> 



\ 



<> 



o' 



Library of Congress Cataloging in Publication Data: 



Jobes, Christopher C. 

Utilizing mechanical linear transducers for the determination of a mining 
machine's position and heading: the concept / by Christopher C. Jobes. 

p. cm. - (Bureau of Mines information circular; 9254) 

Includes bibliographical references. 

Supt. of Docs, no.: I 28.27:9254. 

1. Mining machinery-Automatic control-Data processing. 2. Transducers. 
I. Title. II. Series: Information circular (United States. Bureau of Mines); 9254. 



TN295.U4 



[TN345] 



622 s-dc20 [622\2] 



90-1416 
CIP 



CONTENTS 

Page 

Abstract 1 

Introduction 2 

Global navigation 2 

Local navigation 2 

Face navigation . 3 

Background 3 

Problem definition 3 

Topological requirements 3 

Functional requirements 4 

Constraints 4 

Dimensional constraints 5 

Inertial constraints 5 

Position and heading system 5 

Sensor selection 5 

Sensor configuration 6 

Redundant sensor configuration 6 

Position and heading algorithm 7 

Definition of reference frames 7 

Coordinate transformation of reference frames 8 

Closed-form solution 9 

Loop equation development 9 

Three-transducer solution 9 

Four-transducer solution 11 

Testing and backup strategy 12 

Error analysis 13 

Linear transducers 13 

Analog-to-digital converter 13 

Sample rate 13 

Conclusions 13 

Appendix A.-Linear position transducer calibration data 14 

Appendix B— Calibration equations and error frequency distribution 15 

Appendix C.-Conversion specifications for Intel remote control board 44/20A analog-to-digital converter ... 16 

ILLUSTRATIONS 

1. British Coal trolley-pole-type articulated boom . 4 

2. Instrumented degrees of freedom on trolley-pole-type articulated boom 4 

3. Linear position transducer 6 

4. Kinematic equivalent of linear position transducer 6 

5. Three-transducer configuration 7 

6. Directly solvable redundant four-transducer configuration 7 

7. Local and machine reference frames and position vectors 8 

8. Rectangular and polar coordinate components 8 

9. Configuration for position and heading algorithm 9 

B-l. General calibration equation error frequency distribution 15 





UNIT OF MEASURE ABBREVIATIONS USED IN THIS REPORT 




dB 


decibel 


M v 


microvolt 




°C 


degree Celsius 


/iV/°C 


microvolt per degree Celsius 




ft 


foot 


mV 


millivolt 




ft/s 


foot per second 


oz 


ounce 




Hz 


hertz 


pet 


percent 




in 


inch 


ppm 


part per million 




kHz 


kilohertz 


V 


volt 




kohm 


kilohra 


Vdc 


volt, direct current 




lb 


pound 


V/in 


volt per inch 




US 


microsecond 


7s 


degree per second 



UTILIZING MECHANICAL LINEAR TRANSDUCERS FOR THE 

DETERMINATION OF A MINING MACHINE'S POSITION 

AND HEADING: THE CONCEPT 



By Christopher C. Jobes 1 



ABSTRACT 

This U.S. Bureau of Mines report describes a system to determine the position and heading of a 
mining machine during maneuvers in the face area of an operating mine section. The system is the first 
step in the development of a guidance system for automated mining machines. The position and heading 
algorithm is described, and a preliminary error analysis is performed. 

The sensors employed were linear position transducers mounted on a mining machine with their 
cables attached to points on a stationary reference. Four linear position transducers were used to 
provide a data redundancy that increased the reliability of the guidance system. 

The linear position information obtained from the transducers was processed mathematically via the 
position and heading algorithm to provide the desired position and heading information. The algorithm 
takes advantage of sensor redundancy to continually test the accuracy of the sensor data. 

The major sources of error were determined to be the linear position transducers, the analog-to- 
digital (A/D) converter used to interpret the data, and the sampling frequency of the measurement 
system. These sources of error were evaluated to determine their inaccuracies for use in calculating the 
overall system accuracy. 



'Mechanical engineer, Pittsburgh Research Center, U.S. Bureau of Mines, Pittsburgh, PA. 



INTRODUCTION 



Navigation is one of the key systems that must be de- 
veloped if a mining machine can be automated and the 
machine operator can be moved to a safe, protected area. 
This report describes the approach taken by the U.S. 
Bureau of Mines to solve navigation requirements for 
maneuvering a continuous miner at the coal mine face. 
This work is part of the Bureau's program to enhance 
mine safety. 

The function of a position and heading system for a 
mining machine is to provide guidance information for 
navigation in the face area. This system must be attained 
before face operations can be automated under current 
continuous mining practices. 2 The automation of equip- 
ment at the face will increase health, safety, and efficiency 
in this area. To get a better feel for navigational require- 
ments in mining, it is necessary to briefly look at naviga- 
tion in mines as it is currently performed by workers. 

Navigation in a mine can be divided into three cate- 
gories: global navigation, local navigation, and face nav- 
igation. 3 Each of these navigation categories requires the 
performance of various navigational tasks. 

GLOBAL NAVIGATION 

Global navigation consists of the navigation of a piece 
of mobile mining equipment from one place in a mine to 
another that cannot be seen. This is similar to driving an 
automobile from one city to another. The travel of a man- 
trip is a good example of this form of navigation as it may 
cover several miles. Operators of supply and maintenance 
vehicles also perform this type of navigation on a regular 
basis. The operators of all mobile mining equipment, at 
one time or another, must perform this type of navigation 
to get from the surface to their assigned workplace. Then- 
tasks usually include, among other things, mine mapping, 
path generation, and path following. 

The primary task in global navigation is to move a min- 
ing machine from one place in the mine to another. The 
major resource for accomplishing a navigational task is 
the mine map. A mine map is continually updated as 
changes are made to, or occur in, the mine. These 
changes are usually noted on the mine map as they occur 
since they may affect navigational decisions. Such changes 
may include roof falls, stoppings, dangerous conditions, 
unnavigable portions, etc. 



Schnakenberg, G. H., Jr. Computer-Assisted Continuous Coal 
Mining System-Research Program Overview. BuMines IC 9227, 1989, 
15 pp. 

Anderson, D. L. Framework for Autonomous Navigation of a 
Continuous Mining Machine: Face Navigation. BuMines IC 9214, 1989, 
23 pp. 



Using a mine map, knowledge of the initial position and 
desired destination, and path generation rules, a guidance 
system can usually generate a path from one place in a 
mine to another. The generation of this path may take 
into consideration several items: the normally used path, 
mine conditions along that path, traffic, etc. This form of 
path generation usually determines which entries must be 
traveled and where turns must be made. 

When a guidance system follows a generated path, the 
current position must be continually updated on the mine 
map as well as the generated path. As the m inin g machine 
follows the generated path, it must negotiate entries and 
corners and must also perform collision avoidance, imple- 
mented by manipulating the configuration, position, and 
orientation of the machine. 

LOCAL NAVIGATION 

Local navigation is performed within the area of the 
mine that the guidance system sensors can see. Local 
navigation includes traversing entries and turning crosscuts 
in the performance of some navigational task (e.g., path 
following). The local navigational tasks include scheduling 
as well as the normal mine mapping, path generation, and 
path following functions. 

The local map must be of greater detail than the mine 
map. It should be updated more frequently and should 
include the location of all pieces of mining equipment and 
their function. This information will be of use to the 
scheduling algorithm. 

The movement of one piece of mining equipment from 
one place in the local area to another requires some form 
of scheduling since there is a good amount of activity and 
equipment present in the average continuous mining sec- 
tion. Scheduling is required since there is a cycle of tasks 
to be performed at the face requiring the application of 
several pieces of equipment (continuous miners, shuttle 
cars, roof bolters, rock dusters, etc.) among the faces being 
mined. The performance of these tasks needs to be coor- 
dinated so that there is a minimal amount of interference 
among the tasks performed, thus maximizing the efficiency 
and therefore the productivity of the section. 

The path generation and path following tasks are much 
the same as in global navigation, with only a few added 
complications. If shuttle cars are present in the mining 
section, it may be necessary to avoid their trailing cables 
if they are located in the entry being traveled. Also, there 
is a greater need of care in collision avoidance since there 
are more items hanging from the roof, attached to the ribs, 
or parked in the entries. 



FACE NAVIGATION 

Face navigation is performed by mobile mining equip- 
ment preparing to perform a task in the face area of an 
entry or crosscut development. The type of equipment in 
the face area may include mining equipment such as con- 
tinuous mining machines and roof bolters. The face nav- 
igational tasks include mine mapping, path generation, and 
path following. 

The map of the face area should be of the greatest 
detail and should be updated frequently. The updated 
information should include the volume created by coal 
extraction, the location and types of roof bolts applied in 
the face area, the local geology (which may be determined 
by monitoring the virtual work during the drilling cycle), 
etc. 



The path generation and path following tasks are sim- 
ilar to those of local and global navigation, but include 
some additional tasks. Path generation for coal extraction 
must be in accordance with predetermined mining patterns 
and must develop the entry according to the mine plan. 
Path following must be fairly exact and must function in 
accordance with the type of machine being guided (i.e., the 
cutting drum diameter would determine the distance of 
sump for a continuous mining machine). 

The actual guidance of the mining machine is of great 
importance to the mining cycle since much depends on 
face navigation. This guidance task is important since the 
conditions and obstacles are a hindrance to the implemen- 
tation of a guidance system. 



BACKGROUND 



Many designs of navigational systems for use with 
autonomous self-guided mining machines have been at- 
tempted in prior years. These attempts can be categorized 
into electromagnetic, stress wave, and mechanical methods. 
While there is much work being done in the electromag- 
netic (optical, laser, radar, magnetic compass, natural 
gamma, etc.) and stress wave (ultrasonic, seismic, vibra- 
tion, etc.) areas, very little attention is being paid to me- 
chanical guidance systems. 

The apparent reason for the lack of interest in mechan- 
ical guidance systems is that this method usually involves 
either dead reckoning (e.g., sensing wheel motion) or 
mechanical attachment. In dead reckoning, the errors 
introduced into the system are cumulative, and therefore 
a system relying on this form of navigation alone is unre- 
liable, particularly in a mining environment. Most mining 



machine designers seem unwilling to restrict the motion of 
their machines enough for a mechanical attachment system 
to be a viable option. 

One known attempt has been made at using an attach- 
ment method for the guidance of a mining machine. In 
particular, a trolley-pole-type articulated boom (fig. 1) is 
used to guide a roadheader. 4 This articulated boom has 
six degrees of freedom (fig. 2) that are instrumented (one 
prismatic and five revolute). This method uses standard 
robot kinematics techniques to determine the position and 
orientation of the roadheader with respect to the local 
reference frame. The system performed adequately, but 
was not considered for use since there was not enough 
overhead room available to the continuous mining 
machine. 



PROBLEM DEFINITION 



A system for machine guidance was required to deter- 
mine the position and heading of a mining machine during 
maneuvers required in the face area of an entry. To ad- 
equately design the mechanical system, it was necessary 
to first define the system's requirements and constraints. 

TOPOLOGICAL REQUIREMENTS 

Topological requirements are the minimum set of pa- 
rameters to be defined before the kinematic chain of a 
mechanism can systematically be enumerated. Usually 
this includes the "space" (planar or spatial) in which the 
mechanism moves, the degree of freedom, and either the 
number of links or independent loops needed to obtain a 



finite solution set. At this point in the design procedure, 
any additional specifications should be made to reduce the 
number of solutions even further. 

The nature of motion of a mining machine in a coal 
seam was determined to be spatial rather than planar since 
it cannot be assumed that the coal seam would be abso- 
lutely smooth for even a short distance. Thus, the mining 
machine was determined to have six degrees of freedom 
(x, y, z, roll, pitch, and yaw). 



British Coal, Headquarters Technical Department (Burton on 
Trent, England). Alignment and Profile Guidance of Roadheaders. 
Final report on European Coal and Steel Community Research Project 
7220-AB/810, 1987, 45 pp. 




Figure 1. -British Coal trolley-pole-type articulated boom. 



//////// f* 



KEY 

P 

- r° r3 - Prismatic joint 



- ( ft - Revolute joint 




Roadheader 



The degree of freedom of the system was dependent on 
the number of independent inputs to the system. In this 
case, the mining machine was the input to the guidance 
system and therefore the system had six degrees of free- 
dom. This degree of freedom required that any kinematic 
chain connecting the mining machine to the local refer- 
ence frame have at least six degrees of freedom to work 
properly. 

FUNCTIONAL REQUIREMENTS 

Functional requirements are the tasks to be performed 
by the mechanism. The only task of this mechanism is the 
tracking of the position and orientation of the mining 
machine. In order for the tracking to be accomplished 
without unnecessary redundancy, no more than six degrees 
of freedom were required to connect the mining machine 
to the local reference frame. 

CONSTRAINTS 



Figure 2.-lnstrumented degrees of freedom on trolley-pole- 
type articulated boom. 



When the mechanism was designed, there were two 
types of constraints: dimensional and inertial. These 
constraints determined the feasibility of any mechanisms 
that were found to satisfy the topological and functional 
requirements. 



Dimensional Constraints 

Some of the dimensional constraints on the mechanism 
connecting the mining machine to the local reference 
frame were that the mechanism— 

1. Should perform the required task within the entry 
in which the mining machine is to be working while min- 
imizing contact with the ribs or uncut coal blocks since 
such contact may or may not result in damage to the 
mechanism. 

2. Should minimize its interference with the mining 
machine's task since its purpose is to measure, not to be 
an avoided object. 

3. Should allow up to a 40-ft advance or a 20-ft, 90° 
crosscut to maximize productivity during the automated 
cutting cycle. 

4. Must avoid interference with the mining machine 
conveyor boom and shuttle car (if continuous haulage is 
not being used) since such contact would interfere with 
reliable measurement and may result in damage to the 
mechanism. 



5. Must avoid the ventilation curtain (if present) since 
contact with the curtain may affect one or more of the 
transducers. 

Inertia I Constraints 

Some of the inertial constraints on the mechanism con- 
necting the mining machine to the local reference frame 
were, as follows: 

1. The inertia of the mechanism should be minimized 
to reduce the loading of the mining machine and the 
mechanism's resolving joints. 

2. The mass of the mechanism must be minimized to 
reduce its interference with the operation of the mining 
machine. 

3. The mechanism must be stiff enough to maintain 
accurate measurements during the movements of the min- 
ing machine. 



POSITION AND HEADING SYSTEM 



To develop the position and heading system, the trans- 
ducers to be used were selected first. Next, the config- 
uration of the sensors was addressed so that the accuracy 
and reliability of the position and heading system were 
maximized. Finally, the number of sensors was made 
redundant so that the reliability of the position and head- 
ing system could be continuously monitored. 

SENSOR SELECTION 

In the average coal seam, the headroom available to the 
position and heading system is less than that available to 
the British Coal roadheader guidance system. Therefore, 
an articulated boom was not considered to be the best type 
of connection to make. Aside from the fact that it was a 
cumbersome way to go about measuring the position and 
heading of a mining machine, it would have been big and 
heavy. Because of the weight and length of the system, it 
was possible to foresee a problem with oscillations in the 
boom that would reduce the life of the measurement sys- 
tem and reduce the accuracy of the measurement readings. 
It would be very difficult to make a 40- to 50-ft-long boom 
stiff enough for accurate measurement without also making 
it too big for the job (i.e., cumbersome with a correspond- 
ing lack of mobility). 

Since not all of the degrees of freedom of the links 
connecting the mining machine to the local reference 
frame needed to be instrumented, several groups of links 
(called chains) or their equivalents were used. Further- 
more, while the motion of the mining machine was 



in six degrees of freedom, not all of the motions were 
considered large enough to require measurement. 
(Although coal seams are not "flat," the generally accepted 
change in gradient does not typically exceed 5° over an 
occasional roll between 20 to 50 ft wide within the 
northern Appalachian coal seams.) Therefore, for a 
relatively smooth seam, the z, roll, and pitch measurements 
could be assumed to have a negligible effect on the x, y, 
and yaw measurements (position and heading) with respect 
to other sources of error. If such information was needed, 
it would be possible to use roll and pitch sensors on the 
mining machine to adjust for such errors relative to a 
leveled local reference. Thus, the sensor selection task 
was one of determining which six-degree-of-freedom chain 
could best perform the position and heading 
measurements, but not all of whose degrees of freedom 
need to be instrumented. 

Given that most transducers measure only one degree 
of freedom and few had the necessary linear range, the list 
of possible mechanical attachment measurement schemes 
was reduced to a linear position transducer, or wire pull 
(fig. 3), as it is sometimes called. Although each linear 
position transducer technically has an infinite number of 
degrees of freedom, each wire was modeled as a six- 
degree-of-freedom kinematic chain equivalent (fig. 4). 
Thus, the position and heading system development task 
became a method of configuring linear position trans- 
ducers to perform the position and heading measurement 
task. 





ScaSe, in 

Figure 3.-Linear position transducer. 



R KEY 

- ( - Revolute joint 



-CIZD- Cylindrical joint 




Mining machine 
Figure 4. -Kinematic equivalent of linear position transducer. 

SENSOR CONFIGURATION 

The task of finding a configuration in which the linear 
position transducers could be used to determine the posi- 
tion and heading of a mining machine was relatively sim- 
ple. While it was possible to perform the required meas- 
urement with one linear transducer and two revolute 
transducers to measure the departure angle of the wire 



relative to the local and machine reference frames, there 
was no such transducer on the market. This configuration 
was deemed to be low in accuracy because the linear 
transducers had a relatively light "pull" (about 3 lb), and 
this configuration would not be redundant and therefore 
could be tested for accuracy. Therefore, three linear po- 
sition transducers were deemed required to determine the 
three measured degrees of freedom (position and head- 
ing). The placement of these transducers and the attach- 
ment of their cables made little difference in the ability to 
calculate the position and heading of the mining machine, 
provided the cables were not parallel and the attachment 
points and transducer locations were separated by a suffi- 
cient distance to reduce the linear position transducer 
error to a reasonable percentage. The placement of the 
transducers did, however, affect the computation required 
for the calculation. If the linear position transducers were 
located as shown in the top portion of figure 5, a set of 
nonlinear transcendental equations occurred and an itera- 
tive solution method was required. If the linear position 
transducers were located as shown in the bottom portion 
of figure 5, the solution to the position and heading equa- 
tions could be arrived at easily using elementary trigonom- 
etry. However, because of the nature of the trigonometric 
equations, some ambiguities arose that resulted in multiple 
possible machine positions and headings for a given set of 
sensor readings. 

REDUNDANT SENSOR CONFIGURATION 

An easy method to solve a portion of the multiple so- 
lution problem arising from ambiguities in the trigono- 
metric derivation was to add a redundant sensor (fig. 6). 
This sensor served two purposes. The first purpose was 
to identify which of the multiple solutions was the correct 
one by first performing a very simple four-transducer 
solution. The other purpose was to compare the four 
three-transducer solutions with the four-transducer solu- 
tion to see if one of the four linear position transducers 
was furnishing erroneous data. 

A transducer error would show up as all of the four 
three-transducer solutions being different from each other, 
since only one solution did not use the suspect transducer. 
Of course in this case, the four-transducer solution was 
also incorrect. If more than one transducer was giving 
erroneous data, every three-transducer solution could yield 
a different position and heading. This was, however, pref- 
erable to the case with a single three-transducer solution 
where it was never really known if the solution was correct. 

Using this information, the mining machine could be 
guided using the four-transducer solution. A correct solu- 
tion could still be found, however, if the initial configura- 
tion and the sign of angular rotation of the mining ma- 
chine was known. If a valid solution was not obtainable, 
then the mining machine controller could shut down the 
m inin g machine and request maintenance. 




Reference 






"Wlpt, 


Entry ^-"AA \ 




LPT 2\\ 




Reference 




/\ Mining yk \ 




LPT u^w - ^ 




LPTw r 






V^LPTV^ 





Scale, ft 



KEY 
LPT Linear position transducer 



10 



Scale, ft 



KEY 

LPT Linear position transducer 



Figure 5.-Three-transducer configuration. Top, requiring 
iterative solution; bottom, directly solvable. 



Figure 6.-Directly solvable redundant four-transducer 
configuration. 



POSITION AND HEADING ALGORITHM 



In developing the position and heading algorithm, the 
first task was to define the reference frames used. The 
next task was to develop the coordinate transformation of 
the reference frames. Finally, a closed-form solution of 
the position and heading algorithm was developed. 

DEFINITION OF REFERENCE FRAMES 

In mechanism analysis, the location of an object was of 
some concern. In order to describe an object's position in 
space, a coordinate system or frame was attached to the 
object. The position and orientation of the frame was then 
described with respect to some reference coordinate frame. 
For the purpose of this research, a local reference frame 
was attached to ground at some distance behind the min- 
ing machine. The machine reference frame was attached 
to some point on the mining machine (fig. 7). A reference 
frame was denoted by the prefix superscripts, A, for the 
local reference frame, and B, for the mining machine's 
reference frame. 

It was essential that some notation be defined before 
coordinate transformations could be addressed. The po- 
sition vector was represented by a bold-faced character 
(R). This character represented the position in space of a 
point relative to the associated reference frame. For a 



two-dimensional example, the position vector either de- 
scribed an (x,y) point in rectangular coordinate form or 
an (r,0) point in polar coordinate form (fig. 8). The no- 
tation for the components of a vector in the rectangular 
coordinate system was 



where 



and 



Xj or yj , 
i = vector number, 
x = x component of position vector, 
y = y component of position vector. 



The notation for the components of a vector in the polar 
coordinate system was 



where i = vector number, 

r = magnitude of position vector, 

and 6 = direction of the position vector relative to 

positive x axis. 




Scale, ft 



KEY 

A Local reference 
B Machine reference 
P Point on machine 
R Vector 



Figure 7.-Local and machine reference frames and position 
vectors. 



1 


i 


P(: 


\ 


R,/ 




Yi 






\ 









-«— Xj— - 





P(r,0) 




RECTANGULAR 

COORDINATE 

SYSTEM 



POLAR 

COORDINATE 

SYSTEM 



KEY 
P Point 
R Vector 

Figure 8.-Rectangular and polar coordinate components. 



The relationship between the rectangular and polar coor- 
dinate systems was 

x = r cos(0) and y = r sin(0), 

with the corresponding inverses 

r = (x 2 + y 2 ) 1 / 2 and 6 = tan'^y/x). 

Finally, in matrix notation, the rectangular form was shown 
as 



or 



"r cos(0)l 
_r sin(0)J 



COORDINATE TRANSFORMATION 
OF REFERENCE FRAMES 

Since the mining machine reference frame moved with 
respect to the local reference frame with three degrees 
of freedom, coordinate transformations were required to 
transform position vectors expressed in the mining ma- 
chine coordinate system into position vectors relative to 
the local reference frame. This transformation is standard 
procedure in robot kinematics. The process was almost 
trivial since there were only two reference frames in this 
case. 

Consider again figure 7. If only the position vectors 
A R t and B R 2 were known, elementary vector algebra would 
have dictated that the sum of the two would equal A R 3 . 
That was not true in this case, however, since B R2 was not 
measured in the local coordinate frame. If a transform %T 
was available to convert the position vector from the ma- 
chine coordinate frame into the local coordinate frame by 
a rotation operation of V> degrees, then the resulting equa- 
tion would be 



If 



and 



V R 



^D _ An , ArpBwj 

K 3 ~ K l + B 1 K 2- 

(17,-18) in local reference frame, 



(1) 



R 2 = (20,10) in machine reference frame, 
V> = heading of 20° in local reference frame, 



then it could be shown that 

R 3 = (32,-2) in local reference frame. 
Letting the rotation transform 

cos(V>) -sin(V>)l 
sin(V>) cos(V»)J 



(2) 



and substituting the known values into equation 1 yielded 



[-£] 



sin(20°) cos(20°)_ 



|~cos(20°) -sin(20°)l [20l 



32 
-2 



m + [3 



It should be noted here that 



i 



R,. 



This is the standard method for performing kinematic 
analysis of mechanisms in a plane, which was used during 
the development of the closed-form solution of the posi- 
tion and heading algorithm. 



CLOSED-FORM SOLUTION 

The standard solution method for kinematic problems 
involving more than one reference frame (coordinate 
transformation analysis of a planar mechanism) was ap- 
plied to this position and heading system. While the cho- 
sen configuration of the position and heading system made 
the problem simple enough to handle by elementary trig- 
onometric solution methods, the coordinate transformation 
analysis method was determined to be general enough to 
be applied to all possible configurations of the linear trans- 
ducers, whereas of the infinite number of configurations, 
only a finite number could be addressed utilizing standard 
trigonometric techniques. 

Before a closed-form solution for the position and 
heading algorithm could be performed, the linear position 
transducers were configured for use and all the position 
vectors of interest were defined (fig. 9). The first step in 
developing the position and heading algorithm was to write 
the loop equations for a three-transducer solution and to 
determine if the equations could be reduced into a direct- 
ly solvable form. The next step was to determine the 
closed-form solution. A four-transducer solution was then 
found from the closed-form solutions for the four three- 
transducer solutions. Finally, a strategy was developed in 
which the three-transducer solutions were used to test the 
validity of the four-transducer solution and to provide a 
backup solution should one transducer provide erroneous 
data. 

Loop Equation Development 

Four loop equations were written from the configura- 
tion shown in figure 9, as follows: 



eliminated by subtracting equations 5 and 4 from equation 
3, yielding 



A Rj + A R 4 


ArrBn An 

■ B 1 *9 ." K ll 


= o, 


(3) 


A R X + % 


ArpBwJ Awj 

- gl K 1Q - K n 


= 0, 


(4) 


A R 2 + A R, 


ArpBp An 

■ gl Kg - K n 


= 0, 


(5) 


A R 2 + A R 7 


ArpBn Awj 

■ B 1 K 10" K ll 


= 0. 


(6) 



and 



These four vector loop equations represented eight scalar 
equations and seven unknowns. Thus, the four-transducer 
solution was overconstrained, which was to be expected 
since the fourth transducer was added to provide redun- 
dancy. Thus, the three-transducer solutions are derived 
first using the four combinations of three equations rep- 
resenting six scalar equations and six unknowns. 

Three-Transducer Solution 

The three-transducer solution derived here used vector 
loop equations 3, 4, and 5 (i.e., ignored data from trans- 
ducer 4). To reduce the order of the equations, A R U was 



and 



A R : - A R 2 + A R 4 - A Rg = (7) 

X-^-^T^ + ^Xo-O. (8) 




Entry 



Figure 9.-Configuration for position and heading algorithm. 



10 



Then, by substituting 



K 2 - Kj - K3 


(9) 


and Kq - *^io = ^8> 


(10) 


equations 7 and 8 became 




A R 4 - A K 3 - A K6 =0 


(11) 


and A R 4 - A R5 - ^T B Rg = 0. 


(12) 



Vector equation 11 had two scalar equations and only 
two unknowns (9 4 and 9 6 ) and could, therefore, be solved 
directly in the following manner: 



Letting 



and 



a = 



8 = 3 + n 



(13) 
(14) 



The scalar equations were separated and the trigonometric 
identities cos(a + n) = -cos(a) and sin(a + n) = -sin(a) are 
used. (Hereafter, cd and s0, the kinematic shorthand for 
cos(0) and sin(0), will be used.) 



H<h - 5 )1 = [0 
r 6 s((? 3 - S)J 1.0. 



"r 4 c(a + 6 3 )~\ _ |"r 3 c0 3 
_r 4 s(a + 3 )J [r 3 s9 3 

The x scalar equation component was solved for 8: 

t 3 c9 3 - r 4 c(a + 9 3 ) 



■ (15) 



5 = ±cos" 



'3' 



(16) 



and 5 was substituted into y scalar equation component of 
equation 15 using the identity sinfcos'^x)] = (1 - x 2 )^ 2 , 



r 3 s0 3 = 0. 



(17) 



The square root was isolated on the left side of the equa- 
tion and both sides were squared, yielding 

r 2 , - r 3 c 2 3 + 2r 3 r 4 c0 3 c(a + 9 3 ) - r 2 c 2 (a + 9 3 ) 

= r^s 2 ^ 3 - 2r 3 r 4 s0 3 s(a + 9 3 ) + r 2 s 2 (a + 9 3 ). (18) 

Using the identities sin 2 (0) + cos 2 (0) = 1 and 
cos(a)cos(/3) + sin(a)sin(/3) = cos(a-/8), equation 18 re- 
duced to 



A = A + «4 " ^sH™' 



(19) 



which was essentially an application of the law of cosines 
to a triangle whose sides are known. Thus, since 



±cos 



2,2 2 
r 3 + r 4 " r 6 



2r 3 r 4 



equation 13 was solved to find 



h = 



+ a. 



(20) 



(21) 



Since a is determined by the inverse cosine function, there 
are actually two solutions to equation 21, since cos(-a) = 
cos(a). One must therefore assume that the mining ma- 
chine transducer attachment points will always be to the 
positive x side of the line connecting the transducer cable 
attachment points and therefore take the positive value of 
a. Thus, when 9 4 , 6 5 , 9 6 , and 7 are calculated, the upper 
signs in the equations are to be used. 

Now, since 9 4 is known, equation 12 only had two un- 
knowns (0 5 and ij)) and could be solved in the same man- 
ner as equation 11, letting 



and 1/ = 5 -V> + 7r-0{ 

Separating the scalar equations 

rr 4 c(0 8 + v - ol + M* + h)] 

[r 4 s(9 8 + rf> - e)J Lr 5 s(i/ + 8 )J 



(22) 
(23) 



N-sV-1 kcflgl = 

[SV> C0J L r 8 stf sJ 



(24) 



The x scalar equation component was solved for 1/ 

"r 8 c(0 8 + +) - T A d s + * - 



V = ±cos 



-9* (25) 



and v was substituted into the y scalar equation compo- 
nent of equation 24: 



r 4 s(0 8 + + - + (4 - (r 8 c(0 8 + V) 

- r 4 c(0 8 + V - 0) 2 ) V2 - r 8 s(0 8 + *) = 0. 



(26) 



The square root was isolated on the left side of the equa- 
tion and both sides were squared yielding 

r 2 - r 2 c 2 (0 8 + r(>) + 2r g r 4 c(0 8 + i>)c(9 8 + V - 



-2„2 



2.2/ 



i^(* 8 + v - = n & Xh + v» - 



2„2, 



2r 4 r 8 s(^ 8 + ^)s(0 8 + V - + W(9 8 + j>). (27) 



11 



Equation 27 reduced to 

r 5 = r 4 + r 8 " 2r 4 r 8 ce > 



(28) 



which was essentially an application of the law of cosines 
to a triangle whose sides are known. Thus, since 



r 2 , 2 2 
r 4 + r 8 * r 5 



€ = ±COS 



2r 4 r 8 



(29) 



equation 22 was solved to find the mining machine 
heading. 



V» 4 = ±e 



(The sign to be used for e cannot be determined unless 
both the initial configuration and the sign of the angular 
velocity of the mining machine are known.) 

The position of the mining machine could then be de- 
termined from equation 3 to have components 

r^ii + r r 4 c ^i -Mv^i iv^i =r x iii: od 

therefore, 

x n = r l c6 l + r 4 c0 4 - r 9 c(0 9 + V> 4 ) (32) 

and y n = r l s6 1 + r 4 s0 4 - r 9 s(0 9 + V 4 ). (33) 

In summary, the three-transducer solution, which did 
not use transducer 4 data (i.e., using loop equations 3, 4, 
and 5), for the mining machine's heading was given by 
using equations 20, 21, 29, and 30. The mining machine's 
position was given by equations 32 and 33. 

In a similar manner, in a three-transducer solution, 
which excludes transducer 3 (i.e., using loop equations 3, 
4, and 6), the mining machine's heading is given by 



-l 



and 



7 = ±cos 



e 5 = e 3 ± 7 , 



v = ±cos" 



V>-> = ft - ±u - 



r 2 , 2 2 1 
r 3 + r 5 - r 7 


(34) 


. 2r 3 r 5 . 






(35) 


r 2 , 2 2 l 
r 5 + r 8 * r 4 


(36) 


. 2r 5 r 8 J 





(37) 



The mining machine's position was determined by equation 
4 to have components 

x ll = r i c ^i + r 5 c *5 - r io c (*iO + ^3> (38) 

and y n = t 1 s6 1 + r 5 s0 5 - r lo s(0 lo + V> 3 ). (39) 



Next, in a three-transducer solution, which excludes 
transducer 2 (i.e., using loop equations 3, 5, and 6), the 
mining machine's heading was given by 



(30) and 



_! 


r 2 , _2 2 - 

r 3 + r 6 " r 4 


(40) 


— ±cos 


I 2r 3 r 6 - 




= 6 3 - n T S, 


(41) 


j 


" r 6 + r l - r 7 " 


(4 2 ) 


— ± cos 


- 2r 6 r 8 - 




= ±r,-0 s 


+ *6- 


(43) 



The mining machine's position was determined by equation 
5 to have components 



and 



x n - r 2 c0 2 + r 6 c0 6 - r 9 c(0 9 + V> 2 ) (44) 

y n = r 2 s0 2 + r 6 s0 6 - r 9 s(0 9 + V 2 )- (45) 



Finally, in the last three-transducer solution, which 
excludes transducer 1 (i.e., using loop equations 4, 5, and 
6), the mining machine's heading was given by 



-1 



and 



= ±cos 



= 3 - n T yS, 



f = ±cos" 



V>! = n- ±r- 



r 2 ,2 2i 

r 3 + r 7 r 5 


(46) 


. 2r 3 r 7 . 




: A 


(47) 


" r 7 + r 8 * r 6 ' 


(48) 


- 2r 7 r 8 - 




- e s + e 7 . 


(49) 



The mining machine's position was determined by equation 
4 to have components 

x ll = r 2 c ^2 + x -f- B l - r io c (^io + V>i) (50) 

and y n = r 2 s0 2 + r 7 s0 7 - r lo s(0 lo + ^i)- (51) 

Four-Transducer Solution 

The four-transducer solution was derived through equa- 
tions already derived in the three-transducer solution sec- 
tion. Many viable combinations of these equations could 
perform the desired task, but these combinations relied on 
all four transducers yielding identical results. For the 
purpose of this derivation, equations 20, 21, 46, and 47 
were used. 



12 



With 6 4 and 7 known from the derivations in the pre- 
vious section, the vector loop equation that described this 
four-transducer solution was 

A R X + A R 4 - £1% + bT^IO " AR 7 " AR 2 = °> ( 52 ) 

which by substituting equations 9 and 10, became 



R 



ArrB 



4 " B 



T°Rg - ^R 7 



% = o. 



(53) 



The only unknown in this equation was V>, but to avoid the 
ambiguities that arise from the use of the inverse sine or 
cosine, information from both scalar equations was taken 
into consideration. This was done by isolating the un- 
known quantity on the left side of the equation, dividing 
the y scalar component equation by the x scalar compo- 
nent equation, and solving for the unknown variable, re- 
sulting in 



V> = tan" 1 



TaSVa " T-jSu-j - ToSP-j 



r 4 c^ 4 -r 7 c^ 7 -r 3 c^ 3 . 



(54) 



Of course, to determine the correct heading of the ma- 
chine, the quadrant information yielded by the x and y 
scalar equation components was utilized to eliminate am- 
biguities caused by the inverse tangent function. The 
mining machine's position was calculated in the standard 
manner, yielding 

x n = T i c h + r 4 c ^4 " r 9 c (^9 + V>) (55) 

and y n = t 1 &0 1 + r 4 s0 4 - r 9 s(0 9 + \j>). (56) 

Testing and Backup Strategy 

When measurements of any kind are performed, it is 
usually more important to know the reliability of the data 
than the actual data itself. If the data are not known to be 
reliable, the data are useless, since in mining operations, 
safety of personnel is of extreme importance. Thus, the 
redundancy provided by an additional sensor supplied a 
means of determining the accuracy of the data furnished. 
The four-transducer solution would always yield the cor- 
rect position and heading, provided that the constraints of 
operation were met and the transducers were furnishing 
reliable data. 

In order for the guidance system to operate properly, it 
was assumed that the attachment points of the transducers 
on the mining machine would always remain on the posi- 
tive x side of the imaginary line connecting the transducer 
cable attachment points (fig. 9). If this constraint was not 
met, the heading and position data developed would be 
inaccurate. The best strategy, therefore, was to keep the 
transducer and cable attachment points separated by a 
suitable distance. 

To determine if one of the transducers was furnishing 
unreliable data, a continuous check could be kept by cross 
comparing the four three-transducer solutions. If only one 



transducer was furnishing unreliable data, then the error 
could be determined to exist by the fact that all four so- 
lutions would disagree. If more than one transducer was 
furnishing inaccurate data, then it was possible that no 
error could be identified. While the possibility of more 
than one transducer failing at the same time in a manner 
yielding no identifiable error is small, it was theoretically 
possible in a roof fall or mechanical obstruction of mul- 
tiple cables. The transducer data could be checked in such 
cases for unrealistically large rates of change in length, for 
underrange errors, and for overrange errors. 

Since each three-transducer solution method yielded 
two possible positions and headings, in order for the 
three-transducer solutions to track the four-transducer 
solution, knowledge of the initial configuration of the 
mining machine and the subsequent changes in sign of the 
three-transducer solution equations was required. The 
initial configuration of the mining machine was assumed 
to be similar to that shown in figure 9 and was to be cor- 
roborated by the four-transducer solution. This config- 
uration allowed the initialization of the signs in the head- 
ing equations 



V» 4 = ae 



'4' 



and 

where V\ 



V> 3 = 7r - ai> - 8 + 5 , 

V-2 = b»? - e g + e 6 , 
0i = * - bf - 8 + e 7 , 



(57) 
(58) 
(59) 
(60) 



and 



= heading determined by a three-transducer 
solution, which excludes transducer i, 

= sign of e and v since both change signs at 
same time, 

= sign of r\ and £* since both change signs at 
same time, 



to be such that both a and b were positive in the initial 
position and never change signs at the same time. Chang- 
ing the signs of a and b could be performed by keeping 
track of both headings for each solution and choosing the 
appropriate sign and/or by noting the transition of the 
solution equations and using the sign of the angular veloc- 
ity of the mining machine (u>). Determining the signs of 
a and b using the angular velocity of the mining machine 
was done in the following manner (fig. 9): 

if [(u^O) and (w<0)] or [(u~ir) and (w>0)], then a = -1, 

if [(^=0) and (w>0)] or [(y~ir) and (w<0)], then a = 1, 

if [(»?M)) and (w<0)] or [(»?~7r) and (w>0)], then b = -1, 

and if [(f?~0) and (w>0)] or [(r?~7r) and (w<0)], then b 
= 1. 



13 



Both of these methods relied on information external to 
the three-transducer solutions to determine the signs of 
a and b. If a transducer did fail, the error could be 



determined using equations 57 to 60 instead of equations 
30, 37, 43, and 49. 



ERROR ANALYSIS 



Once the position and heading algorithm had been 
designed to reduce the possibility of mathematical errors 
induced by the configuration of the position and heading 
system, it was necessary to determine the source of other 
possible errors. These errors were introduced into the 
position and heading calculation through the inherent 
inaccuracies in the linear position transducers, the analog- 
to-digital (A/D) converter, and the sample rate. 

The effects of these errors on the position and heading 
calculation depended to a large extent on the position and 
heading of the mining machine. While the determination 
of the maximum error required either an exhaustive trig- 
onometric proof involving the taking of the partial deriv- 
atives of the governing equations or a numerical methods 
analysis, a preliminary numerical analysis utilizing a com- 
puter model of the position and heading system indicated 
that the overall position and heading system error was on 
the order of 0.25 ft and 3°, respectively. While error prop- 
agation could not be easily derived, the errors introduced, 
however, could be quantized. 

LINEAR TRANSDUCERS 

Linear transducers of the length necessary to perform 
the necessary tasks in the position and heading system 
(approximately 750 in) had a stated accuracy of 0.1 pet full 
scale, but typically had an accuracy of 0.05 pet full scale. 
Factory calibration information on five Rayelco 5 P-750A 
linear position transducers selected for use in the position 
and heading system can be found in appendix A. A pre- 
liminary analysis was performed on these calibration data 
to determine the calibration equations for each linear 
position transducer and for all five combined. The fre- 
quency distribution of error from the combined calibration 
equation was determined and appeared to be Gaussian in 



nature. (The calibration equations and error frequency 
distribution can be found in appendix B.) The only other 
accuracy concern for the linear position transducers was 
the effect of the dust and accumulations of dust on the 
transducer's wire itself. If this was discovered to be a 
problem, a submersible type of linear transducer could be 
used with water to flush the interior of the cable takeup 
housing. 

ANALOG-TO-DIGITAL CONVERTER 

The A/D conversion circuitry on the Intel remote con- 
trol board (iRCB) 44/20A used to implement the position 
and heading system had an accuracy of about 0.035 pet full 
scale. These errors were due in part to nonlinearity, inher- 
ent quantizing errors, gain error, zero error, noise, and 
sample and hold dynamic error, and were found in the 
iRCB 44/20A's hardware user's manual. The specifica- 
tions for the A/D converter can be found in appendix C. 

SAMPLE RATE 

Because of the processing capability available to the 
position and heading system through the Intel 8051 micro- 
processor running in a distributed control executive (iDCX 
51) real-time multitasking environment, the number of 
position and heading calculations performed was on the 
order of 5 Hz. Considering, for example, that a Joy 16CM 
mining machine can attain speeds of up to 0.54 ft/s and 
angular velocities of up to 3.23°/s, 6 the sampling errors 
incurred of 0.1 ft and 0.6° could be considered acceptable 
in a mining environment. If greater accuracy is desired, 
a slower speed could be used or the machine could be 
stopped altogether. 



CONCLUSIONS 



A system to determine the position and heading of a 
mining machine during maneuvers required in the face 
area of an operating mine section was described. This 
system utilized four commercially available linear position 
transducers to obtain the required sensory data. The sys- 
tem employed sensor redundancy in its sensor fusion 



scenario to increase the system reliability. The derivation 
of the position and heading algorithm was presented and 
a preliminary error analysis was performed, which showed 
that the position and heading system had sufficient ac- 
curacy for mining application. 



Reference to specific products does not imply endorsement by the 
U.S. Bureau of Mines. 



Sammarco, J. J. Closed Loop Control for a Continuous Mining 
Machine. BuMines RI 9209, 1988, 17 pp. 



14 



APPENDIX A.-LINEAR POSITION TRANSDUCER CALIBRATION DATA 



(Range, 750 in; cable tension, 24 oz; potential resistance, 500 ohms; excitation, 10 V dc) 



Calibration 
step 



Travel, 
in 



Output, 
V 



Ideal, 
V 



Delta, 
mV 



TRANSDUCER 1 



1 0.00 

2 187.50 

3 357.00 

4 562.50 

5 750.00 

1 0.00 

2 187.50 

3 357.00 

4 562.50 

5 750.00 

1 0.00 

2 187.50 

3 357.00 

4 562.50 

5 750.00 

1 0.00 

2 187.50 

3 357.00 

4 562.50 

5 750.00 

1 0.00 

2 187.50 

3 357.00 

4 562.50 

5 750.00 

'0.05 pet full scale, 1.29 mV/(V/in) position. 
^.OS pet full scale, 1.30 mV/(V/in) position. 
3 0.07 pet full scale, 1.29 mV/(V/in) position. 
4 0.04 pet full scale, 1.30 mV/(V/in) position. 



0.035 
2.463 
4.889 
7.314 
9.747 



0.035 
2.463 
4.891 
7.319 
9.747 



TRANSDUCER 2 2 



0.028 
2.462 
4.891 
7.321 
9.758 



0.028 
2.460 
4.893 
7.325 
9.758 



TRANSDUCER 3 3 



0.034 
2.454 
4.887 
7.314 
9.743 



0.034 
2.461 
4.888 
7.316 
9.743 



TRANSDUCER 4 4 



0.030 
2.461 
4.898 
7.339 
9.771 



0.030 
2.465 
4.901 
7.336 
9.771 



TRANSDUCER 5 3 



0.026 
2.448 
4.877 
7.298 
9.731 



0.026 
2.452 
4.878 
7.305 
9.731 



15 



APPENDIX B.-CALIBRATION EQUATIONS AND ERROR FREQUENCY 

DISTRIBUTION 



Output (Qo), in volts, for the five linear position trans- q . 0.02780 

ducers, as described in appendix A, was used to calculate Qi = ± 3.201727 in, 

the slopes, intercepts, and standard deviations of the cali- 0.012992 
bration equations for each of the five linear position trans- 
ducers and for all five combined. The general calibration with maximum error = 0.246305 in. 
equation was determined to be 

The calibration equation for transducer 5 was determined 

Qo - 0.02944 t0 be 

Qi = ± 2.050786 in, 

°- 012959 Qo - 0.02580 

Qi = ± 2.772926 in, 



with maximum error = 1.694514 in. 



0.012940 



The calibration equation for transducer 1 was determined with maximum error = 0.030911 in. 
to be 



Qo - 0.03460 

Qi = ± 0.89625 in, 

0.012947 

with maximum error = 0.239444 in. 



The error frequency distribution for the general calibration 
equation is as shown in figure B-l. 



The calibration equation for transducer 2 was determined 
to be 

Qo - 0.02820 

Qi = ± 1.014279 in, 

0.012970 

with maximum error = 0.223591 in. 

The calibration equation for transducer 3 was determined 
to be 

Qo - 0.03080 

Qi = ± 1.120895 in, 

0.012948 

with maximum error = 0.35526 in. 

The calibration equation for transducer 4 was determined 
to be 



6 
5 
4h 



> 

o 

z 

lit 

=> 3 

o 

LlI 

<r 

"■ 2 



1 1 ' 1 ' 1 


v\ 


1 
— > 


ft 


i ■ i ■ i ■ 


/ 




V\ 


* 


\ 






/ 


/ 








/ 


/ 






/. 


/ 


/ 


V 








$ 


* 


', 




— -T i 


1 


1 


1 


1 




: 


I 




I 


i 


1 



-2.0 -1.5 -1.0 



-0.5 0.5 

ERROR, in 



1.0 



1 .5 2.0 



Figure B-1 .-General calibration equation error frequency 
distribution. 



16 



APPENDIX C.-CONVERSION SPECIFICATIONS FOR INTEL REMOTE 
CONTROL BOARD 44/20A ANALOG-TO-DIGITAL CONVERTER 



Linearity 

Differential linearity 

Inherent quantizing error 

System accuracy: 

Gain = 1 

Gain = 10 

Gain = 100 

Gain = 500 

Gain error 

Zero error 

Channel crosstalk 

Noise (A/D converter) 

Instrumentation amplifier settling time: 

Gain = 1 

Gain = 10 

Gain = 100 

Gain = 500 

A/D conversion time: All gains 

Maximum A/D throughput: 

Gain = 1 

Gain = 10 

Gain = 100 

Gain = 500 

Sample and hold feedthrough attenuation 

Sample and hold dynamic error 

Offset drift (gain = 1) 

Input offset drift multiplied by gain 

Gain drift: 

Gain = 1 

Gain = 10 

Gain = 100 

Gain = 500 

Monotonocity 

Common mode rejection ratio: 

Gain = 1 

Gain = 500 

Amplifier input noise 

Channel-to-channel input voltage error . . 

Resolution 

FSR Full-scale reading. 
LSB Least significant bits. 
RMS Root mean square. 



To within ±0.75 LSB. 
To within ±0.75 LSB. 
±0.5 LSB. 

To within ±0.035 pet FSR. 
To within ±0.05 pet FSR. 
To within ±0.07 pet FSR. 
To within ±0.15 pet FSR. 
Adjustable to zero. 

Do. 
-80 dB at 1 kHz. 
0.2 LSB RMS. 

20 /is. 
20 lis. 
100 ms. 
100 its. 
30 lis. 

20,000 samples per second. 

Do. 
7,500 samples per second. 

Do. 
-80 dB at 1 kHz. 
±0.75 LSB. 
100mV/°C. 
3 mV/°C. 

32 ppm of FSR per degree Celsius. 
40 ppm of FSR per degree Celsius. 
65 ppm of FSR per degree Celsius. 
75 ppm of FSR per degree Celsius. 
Monolithic, 0° to +60° C. 

70 dB at 60 Hz, 1 kohm unbalance. 
100 dB at 60 Hz, 1 kohm unbalance. 
2 ilV RMS. 
±40 /iV. 
12 bits. 



INT.BU.OF MINES,PGH.,PA 29151 



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